Complex Conjugate
Functions & Advanced AlgebraThe complex conjugate of a + bi is a - bi, formed by negating the imaginary part while keeping the real part unchanged.
Formula
\overline{a + bi} = a - bi
Definition
The complex conjugate of $a + bi$ is $a - bi$: just flip the sign of the imaginary part. It is the mirror image of the complex number across the real axis.
Example
The conjugate of $3 + 5i$ is $3 - 5i$. The conjugate of $2 - 4i$ is $2 + 4i$. The conjugate of $6$ is just $6$ (since $6 = 6 + 0i$).
Key Insight
Multiplying a complex number by its conjugate always gives a real number: $(a+bi)(a-bi) = a^2 + b^2$. This trick is used to divide complex numbers by "rationalizing" the denominator.
Definition
The conjugate of $z = a + bi$ is $\bar{z} = a - bi$. Key properties: $z\bar{z} = a^2 + b^2 = |z|^2$ (always real and non-negative). $z + \bar{z} = 2a = 2\text{Re}(z)$. $z - \bar{z} = 2bi$. Used to divide: $(a+bi)/(c+di) = (a+bi)(c-di) / (c^2+d^2)$.
Example
Divide $(2+3i)/(1-2i)$: multiply numerator and denominator by conjugate $(1+2i)$: $(2+3i)(1+2i)/((1-2i)(1+2i)) = (2+4i+3i+6i^2)/(1+4) = (2+7i-6)/5 = (-4+7i)/5 = -4/5 + (7/5)i$.
Key Insight
Complex roots of polynomials with real coefficients always come in conjugate pairs. If $a + bi$ is a root, so is $a - bi$. This is why quadratic equations have either two real roots or two complex conjugate roots.
Definition
Complex conjugation $z \to \bar{z}$ is the unique non-trivial field automorphism of $\mathbb{C}$ fixing $\mathbb{R}$. It is an involution ($\bar{\bar{z}} = z$) and satisfies $\overline{z+w} = \bar{z} + \bar{w}$ and $\overline{zw} = \bar{z}\bar{w}$. In Hermitian inner product spaces, conjugation appears in the definition: $\langle z, w \rangle = \bar{z} w$ (or its generalization).
Example
The Hermitian transpose (conjugate transpose) of a matrix $A$ is $A^* = \bar{A}^T$. A matrix $A$ is Hermitian if $A^* = A$, and all Hermitian matrices have real eigenvalues. This generalizes symmetric matrices to complex vector spaces.
Key Insight
Conjugation is the automorphism that distinguishes $\mathbb{C}$ from its "mirror image." The insistence that physical observables correspond to Hermitian operators in quantum mechanics is essentially the requirement that measurement outcomes (eigenvalues) are real, enforced via conjugate symmetry.